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arXiv · 1902.05593

Antipodal Hadwiger numbers of finite-dimensional Banach spaces

S. K. Mercourakis·G. Vassiliadis
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Abstract

Let $X$ be a finite-dimensional Banach space; we introduce and investigate a natural generalization of the concepts of Hadwiger number $H(X)$ and strict Hadwiger number $H'(X)$. More precisely, we define the antipodal Hadwiger number $H_α(X)$ as the largest cardinality of a subset $S \subseteq S_X$, such that $\forall x \neq y \in S \,\,\, \exists f \in B_{X^*}$ with \[1 \le f(x)-f(y) \,\,\, \textrm{and} \,\,\, f(y) \le f(z) \le f(x) \,\,\, \textrm{for} \,\,\, z \in S.\] The strict antipodal Hadwiger number $H'_α(X)$ is defined analogously. We prove that $H'_α(X)=4$ for every Minkowski plane and estimate (or in some cases compute) the numbers $H_α(X)$ and $H'_α(X)$, where $X=\ell_p^n, 1 < p \le +\infty$ and $n \ge 2$. We also show that the number $H'_α(X)$ grows exponentially in $\dim X$.

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S. K. Mercourakis, G. Vassiliadis. 2020-06-12. Antipodal Hadwiger numbers of finite-dimensional Banach spaces. https://doi.org/10.1007/s13366-020-00510-x

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